Semi-classical measures

Moreover, in both cases we have

Nlim→∞

1

N Tr[γ_N] = Z

R³

ρ(x) dx= 1,

and the densities _N¹ρ_γN converge to ρ weakly in L¹(R³) and inL^5/3(R³). The same conclusions also hold if γ_N is replaced by the projectionγ˜_N onto theN lowest eigen- vectors of the operator used to define γN.

Proof. The proof of Lemma A.24 also holds mutatis mutandis for this lemma, and we omit the details.

Using the trial states constructed above we can now show the upper bound on the energy.

Proof of Proposition A.22.. Let 0 ≤ ρ ∈ Cc(R³) with R

ρ(x) dx = 1, and take ˜γN

as in either Lemma A.24 or Lemma A.25, depending on the sequence (β_N). Since V ∈L^5/2_loc(R³) and ρ_˜γN is supported inside the box C_R, we get by weak convergence of _N¹ρ_γ˜N that

1 N

Z

R³

V(x)ρ˜γN(x) dx−→

Z

CR

V(x)ρ(x) dx

as N tends to infinity. By the Stone-Weierstrass theorem, we may approximate w(x−y)inL⁵²(C_R²)by a function of the formw₀=Pk

j=1g_j⊗h_j withg_j, h_j ∈C(C_R).

By a standard approximation argument we conclude that 1

N² Z Z

R⁶

w(x−y)ρ_γ˜N(x)ρ_γ˜N(y) dxdy −→

Z Z

C_R²

w(x−y)ρ(x)ρ(y) dxdy.

Hence, continuing from (A.52) withγ_Ψ= ˜γ_N, we find (for instance in the case where β_N →β ∈(0,∞))

lim sup

N→∞

E(N, β_N)

N ≤ E_β^MTF(ρ).

If β_N →0, orβ_N → ∞, we obtain analogous bounds by appealing to Lemma A.25.

This concludes the proof since the Thomas-Fermi ground state energy can be ob- tained by minimizing over compactly supported, continuous functions, and ρ ∈ C_c(R³)is arbitrary.

limit. The constructions in this section are only useful for dealing with the case whereβN →β >0. In the case whereβN →0, it is more convenient to use the same semi-classical measures as in [5]. This case is treated in Section A.6.

The first step is to diagonalize the three-dimensional magnetic Laplacian, i.e., we consider

H_A= (−i∇+A)² =H_A⊥−∂_x²₃, (A.63) where A^⊥(x1, x2) = ¹₂(−x₂, x1), andH_A^⊥ := (−i∇+A^⊥)² acts on L²(R²). Letting F₂ denote the partial Fourier transform in the second variable on L²(R²), and T the unitary operator on L²(R²)defined by(T ϕ)(x₁, ξ) =ϕ(x₁+ξ, ξ), an elementary calculation shows that

H_A^⊥e^i12x1x2F₂⁻¹T =e^i12x1x2F₂⁻¹T

− d²

dx²₁ +x²₁

⊗1L²(R)

. (A.64)

It is very well known that the harmonic oscillator admits an orthonormal basis of eigenfunctions (fj)j≥0 for L²(R), with (−_dx^d22 +x²)fj = (2j + 1)fj and f0(x) = π⁻¹⁴e^−12x2. In particular, equation (A.64) means for anyj ≥ 0 and any normalized Schwartz function v on R, thate^i12x1x2F₂⁻¹T(f_j⊗v) is a normalized eigenfunction for H_A^⊥ with corresponding eigenvalue2j+ 1.

Suppose that ϕ is an eigenfunction for H_A^⊥ corresponding to 2j + 1. If we scale the magnetic field and instead consider H_BA^⊥ = (−i∇+BA^⊥)², and denote x×y = x1y2 −x2y1 for x, y ∈ R², we see for any fixed y ∈ R² that ϕey,B(x) :=

√Be^−iB2y×xu_j(√

B(x−y))is an eigenfunction forH_BA⊥ corresponding to the eigen- valueB(2j+ 1).

A.4.1 Coherent states

Throughout this subsection,~and bwill denote arbitrary positive numbers, that is, the scaling relations (A.5) will not be needed. Forf ∈L²(R³) we denote by f^~ the function

f^~(y) =~⁻

3 4f(~⁻

1 2y).

Definition A.26. We fix a normalized f ∈L²(R³), and for eachj ∈N0 we choose any normalized eigenfunction ϕ_j in thej’th Landau level of H_A^⊥. For fixedx∈R², u∈R³,p∈R, and~, b >0, we define functionsϕ^~,b_x,j onR² andf_x,u,p,j^~,b onR³ by

ϕ^~_x,j^,b(y⊥) =~⁻

1

2b¹²e^{−i2b~x×y⊥}ϕ_j(~⁻

1

2b¹²(y⊥−x)), (A.65) and

f_x,u,p,j^~,b (y) =ϕ^~_x,j^,b(y⊥)f^~(y−u)e^ipy~3, (A.66) wherey⊥ = (y₁, y₂)denotes the part of y orthogonal to the magnetic field.

Note thatϕ^~_x,j^,b is an eigenfunction for the operatorH_~⁻¹_bA^⊥ corresponding to the eigenvalue ~⁻¹b(2j+ 1). Later we will put further assumptions on the function f, but for now it can be any normalized L²-function.

We will use (A.65) and (A.66) to build Landau level projections and some reso- lutions of the identity. Recall that for any normalized functionv∈L²(R) we have a resolution of the identity

1 2π

Z

R²

|v_x,pihv_x,p|dxdp=1L²(R), (A.67) where v_x,p(y) =v(y−x)e^ipy,x, p∈R. We will use shortly that ifu ∈L²(R) is any other function, then

1 2π

Z

R²

hψ, v_x,pihu_x,p, ψidxdp

= 1 2π

Z

R²

hu, ψ_−x,−pihψ_−x,−p, vidxdp=hu, vikψk². (A.68) Lemma A.27. Let Π⁽²⁾_j denote the projection onto the j’th Landau level of the op- erator H_~⁻¹_bA^⊥. We have that

b 2π~

∞

X

j=0

Z

R²

|ϕ^~_x,j^,bihϕ^~_x,j^,b|dx=1L²(R²), (A.69) and

b 2π~

Z

R²

|ϕ^~_x,j^,bihϕ^~_x,j^,b|dx= Π⁽²⁾_j . (A.70) Proof. For ϕ ∈ L²(R²) we denote ϕe_x(y) = e^−2ix×yϕ(y−x), and recalling the iso- morphism (A.64), we furthermore define a unitary operator U := T^∗F₂e^{−i12(·)1(·)2}. Utilizing the usual properties of the Fourier transform, we have for any function ϕ that

F₂

e^{−i2(·)1(·)2}ϕex

(y1, ξ)

=e^−2ix1x2F₂

e^{−2i((·)1+x1)((·)2−x2)}ϕ(· −x) (y₁, ξ)

=e^−2ix1x2e^−ix2ξF₂

e^{−2i((·)1+x1)(·)2}ϕ((·)₁−x₁,(·)₂) (y₁, ξ)

=e^−2ix1x2e^−ix2ξF₂

e^{−2i(·)1(·)2}ϕ

(y₁−x₁, ξ+x₁), and so

(Uϕex)(y1, ξ) =e^−2ix1x2e^−ix2ξ(U ϕ)(y1, ξ+x1). (A.71) Introducing the parameterα:=b/~and denoting byVαthe unitary operator given by (Vαψ)(y) :=√

αψ(√

αy), thenϕ^~_x,j^,b(y) = (Vαϕ^1,1√_αx,j)(y). SinceU ϕj is an eigenvector

corresponding to the j’th eigenvalue of (− ^d2

dx²₁ +x²₁)⊗1L²(R), we can write U ϕ_j =

∞

X

k=1

c_kf_j⊗v_k, where(v_k)is any orthonormal basis ofL²(R), andP

k|c_k|²= 1. Combining this with (A.71) and using (A.68), we obtain

b 2π~

Z

R²

ϕ^~_−x,j^,b , ψ

2dx

= α 2π

Z

R²

U ϕ_j((·)₁,(·)₂−√ αx₁)eⁱ

√αx2(·)2, U V_α^∗ψ

2dx

= 1 2π

Z

R²

X

`,k

c`ck

fj⊗(v`)x1,x2, U V_α^∗ψ

U V_α^∗ψ, fj⊗(vk)x1,x2

dx

=X

`,k

c`ck

U V_α^∗ψ,(|f_jihf_j| ⊗ hv_`, vki1L²(R))U V_α^∗ψ

=

U V_α^∗ψ,(|f_jihf_j| ⊗1L²(R))U V_α^∗ψ .

This actually shows (A.70), sinceΠ⁽²⁾_j =VαU^∗(|f_jihf_j| ⊗1L²(R))U V_α^∗ by the unitary equivalence (A.64). Summing over allj, we also get

b 2π~

∞

X

j=0

Z

R²

ϕ^~,b_x,j, ψ

2dx=kU V_α^∗ψk²=kψk², concluding the proof.

Definition A.28. Using the functions from (A.66), we define operators onL²(R³) by

P_u,p,j^~,b :=

Z

R²

|f_x,u,p,j^~,b ihf_x,u,p,j^~,b |dx. (A.72) Applying the lemma above and using (A.67), it is easy to show the following Lemma A.29. The P_u,p,j^~,b yield a resolution of the identity on L²(R³), i.e.,

b (2π~)²

∞

X

j=0

Z

R

Z

R³

P_u,p,j^~,b dudp=1L²(R³). Furthermore, P_u,p,j^~,b is a trace class operator withTr(P_u,p,j^~,b ) = 1.

Proof. Recall that for y ∈R³ we denote byy⊥ = (y₁, y₂)∈R² the coordinates of y orthogonal to the magnetic field. Let ψ∈L²(R³) and define an auxiliary function

g_u,p^~ (y⊥) :=

f^~(y⊥−u⊥, · −u₃)e^ip~(·), ψ(y⊥, ·)

L²(R)

=√ 2πF₃

f^~(· −u)ψ y⊥,p

~

, (A.73)

withF₃being the partial Fourier transform in the third variable. Using LemmaA.27, we calculate

hψ, P_u,p,j^~,b ψi= Z

R²

f_x,u,p,j^~,b , ψ

2dx

= Z

R²

ϕ^~,b_x,j, g_u,p^~

2dx= 2π~

b

g^~_u,p,Π⁽²⁾_j g_u,p^~

, (A.74)

implying that b (2π~)²

∞

X

j=0

Z

R

Z

R³

ψ, P_u,p,j^~,b ψ dudp

= 1 2π~

∞

X

j=0

Z

R

Z

R³

g_u,p^~ ,Π⁽²⁾_j g_u,p^~ dudp

= 1

~ Z

R

Z

R³

Z

R²

F₃

f^~(· −u)ψ y⊥,p

~

2dy⊥dudp

= Z

R³

Z

R³

|f^~(y−u)ψ(y)|²dydu=hψ, ψi.

To calculate the trace of P_u,p,j^~,b , we take an arbitrary orthonormal basis (ψ_`) of L²(R³) and use the definition of the coherent states (A.65) and (A.66)

Tr(P_u,p,j^~,b ) =

∞

X

`=1

ψ`, P<sub>u,p,j</sub><sup>~,b</sup> ψ`

= Z

R²

f_x,u,p,j^~,b

2 2dx

= Z

R²

Z

R³

b

~

|ϕ_j(~⁻¹²b¹²(y⊥−x))|²|f^~(y−u)|²dydx= 1.

A.4.2 Semi-classical measures on phase space

Let P_±1 denote the projections onto the spin-up and spin-down components in C², that is,

P₁= 1 0

0 0

, P−1= 0 0

0 1

.

We recall that the phase space is Ω = R³×R×N0× {±1}, and that we use the notational convention (A.9). We define k-particle semi-classical measures as follows.

Definition A.30.For ΨN ∈ VN

L²(R³;C²) normalized and 1 ≤ k ≤ N, the k- particle semi-classical measure on Ω^k is the measure with density

m^(k)_f,Ψ

N(ξ) = N!

(N −k)!

D

Ψ_N,O^k

`=1

P_u^~,b

`,p`,j`P<sub>s`</sub>

⊗1N−kΨ_NE

L²(R^3N;C^2N), where1N−k is the identity acting on the lastN −k components ofΨ_N.

The semi-classical measures have the following basic properties. The upper bound in (A.75) below is a manifestation of the Pauli exclusion principle.

Lemma A.31. The function m^(k)_f,Ψ

N is symmetric on Ω^k and satisfies 0≤m^(k)_f,Ψ

N ≤1, (A.75)

b^k (2π~)^2k

Z

Ω^k

m^(k)_f,Ψ

N(ξ) dξ= N!

(N −k)!, (A.76)

and for k≥2, b (2π~)²

Z

Ω

m^(k)_f,Ψ

N(ξ₁, . . . , ξ_k) dξ_k= (N−k+ 1)m^(k−1)_f,Ψ

N(ξ₁, . . . , ξk−1). (A.77) Proof. We will start out by proving (A.75), and we will concentrate on the case k = 1, since the proof easily generalizes to k ≥ 2. Note that 0 ≤ m^(k)_f,Ψ

N obviously holds, as the P_u,p,j^~,b ’s are positive operators. Since P_u,p,j^~,b is trace class, we may write P_u,p,j^~,b =P

kλ_k|ψ_kihψ_k|, where theψ_kconstitute an orthonormal basis ofL²(R³), and P

kλ_k= Tr(P_u,p,j^~,b ) = 1. Note that for any ψ∈L²(R³) we can rewrite, as operators acting on VN

L²(R³;C²),

N(|ψihψ|P_s⊗1N−1) =

N

X

k=1

1k−1⊗ |ψihψ|P_s⊗1N−k, where

X^N

k=1

1k−1⊗ |ψihψ|P_s⊗1N−k

2

=

N

X

k=1

1k−1⊗ |ψihψ|P_s⊗1N−k

+ 2 X

1≤k<`≤N

1k−1⊗ |ψihψ|P_s⊗1`−k−1⊗ |ψihψ|P<sub>s</sub>⊗1N−`.

Each term in the last sum acts as zero on anti-symmetric functions, implying for any ψ∈L²(R³) thatN|ψihψ|P_s⊗1N−1 is an orthogonal projection onVNL²(R³;C²).

We arrive at the conclusion that m⁽¹⁾_f,Ψ

N(u, p, j, s) =

∞

X

k=0

λ_kN

Ψ_N,(|ψ_kihψ_k|P_s⊗1N−1)Ψ_N

≤1.

The result for general k follows by applying what we have just shown k times, so (A.75) holds.

The compatibility relation (A.77) follows by applying LemmaA.29:

b (2π~)²

Z

Ω

m^(k)_f,Ψ

N(ξ₁, . . . , ξ_k) dξ_k

= N! (N−k)!

X

sk=±1

D ΨN,

O^k−1

`=1

P_u^~,b

`,p`,j`P<sub>s`</sub>

⊗ P_sk⊗1N−kΨN

E

= (N −k+ 1)m^(k−1)_f,Ψ

N(ξ₁, . . . , ξ_k−1).

Finally, (A.76) is obtained by repeating thisk−1 more times.

The next two lemmas assert some particularly nice properties of the semi-classical measures, which will prove to be of great importance later. The first one states that the position densities of the measures are like the position densities (A.21) of the wave function Ψ_N.

Lemma A.32 (Position densities). Let Ψ ∈ VNL²(R³;C²) be any normalized wave function, and suppose that that f is a real, L²-normalized and even function, we have for 1≤k≤N that

b^k (2π~)^2k

X

j∈(N0)^k

Z

R^k

m^(k)_f,Ψ(u, p, j, s) dp=k! ρe_Ψ^(k)∗(|f^~|²)^⊗k

(u, s), (A.78)

where the convolution in the right hand side is the ordinary position space convolution in each spin component of ρe_Ψ^(k).

Proof. For notational convenience we introduce an arbitrary Φ∈L²(R^3N). Think of Φ as being one of the spin components ofΨ. Note first that

P_u^~,b

1,p1,j1 ⊗ · · · ⊗P_u^~,b

k,pk,jk = Z

R^2k

|⊗^k_`=1f_x^~,b

`,u`,p`,j`ih⊗^k_`=1f_x^~,b

`,u`,p`,j`|dx, and that for each fixed y∈R^3(N−k) we have as in (A.73) that

⊗^k_`=1f_x^~,b

`,u`,p`,j`,Φ(·, y)

L²(R^3k)

= (2π)^k2

⊗^k_`=1ϕ^~_x^,b

`,j`,F₃^⊗k

(f^~)^⊗k(· −u)Φ(·, y) (·,~⁻

1 2p)

L²(R^2k). Combining these observations and using Lemma A.27, we get

1 (2π)^k

X

j∈(N0)^k

Z

R^k

Φ,(P_u^~,b_1,p1,j1⊗ · · · ⊗P_u^~,b

k,pk,jk)⊗1N−kΦ dp

= 1

(2π)^k X

j∈(N0)^k

Z

R^k

Z

R^3(N−k)

Z

R^2k

⊗^k_`=1f_x^~,b

`,u`,p`,j`,Φ(·, y)

2dxdydp

= (2π~)^k b^k

Z

R^k

Z

R^3(N−k)

F₃^⊗k

(f^~)^⊗k(· −u)Φ(·, y) (·,~⁻

1 2p)

2

L²(R^2k)dydp

= (2π)^k~^2k b^k

Z

R^3(N−k)

(f^~)^⊗k(· −u)Φ(·, y)

2

L²(R^3k)dy.

Applying this toΨ and using thatf is even, we obtain X

j∈(N0)^k

Z

R^k

m^(k)_f,Ψ(u, p, j, s) dp

= X

r∈{±1}^N

X

j∈(N0)^k

Z

R^k

N! (N −k)!

D

Ψ(·;r), O^k−1

`=1

P_u^~,b

`,p`,j`P<sub>s`</sub>

Ψ(·;r) E

dp

= (2π~)^2kN!

b^k(N−k)!

X

r∈{±1}^N−k

Z

R^3(N−k)

(f^~)^⊗k(· −u)Ψ(·, y;s, r)

2

L²(R^3k)dy

= (2π~)^2k

b^k k! ρe_Ψ^(k)∗(|f^~|²)^⊗k (u, s), concluding the proof.

Lemma A.33 (Kinetic energy). Let Ψ∈VN

L²(R³;C²) be normalized, and sup- pose that f ∈C_c^∞(R³) is real-valued,L²-normalized and even. Then we have

D Ψ,

N

X

j=1

(σ·(−i~∇_j+bA(x_j)))²ΨE

=−~N Z

R³

(∇f(u))²du

+ b

(2π~)² X

s=±1

∞

X

j=0

Z

R

Z

R³

(p²+~b(2j+ 1 +s))m⁽¹⁾_f,Ψ(u, p, j, s) dudp. (A.79)

Proof. The assumption thatf is both smooth and compactly supported is far from optimal, but it will be sufficient for our purposes. The assertion of the lemma will hold as long as f^~ satisfies the following version of the IMS localization formula [20, equation (3.18)]

ψ, f^~(−i~∇+bA)²f^~ψ

=

ψ,(f^~)²(−i~∇+bA)²ψ +~²

ψ,(∇f^~)²ψ

(A.80) for any ψ in the domain of (−i~∇+bA)². Since f is normalized, the IMS formula yields

ψ,(−i~∇+bA)²ψ

= Z

R³

ψ, f^~(· −u)²(−i~∇+bA)²ψ du

= Z

R³

ψ, f^~(· −u)(−i~∇+bA)²f^~(· −u)ψ du

−~kψk²₂ Z

R³

(∇f(u))²du. (A.81)

Returning to the semi-classical measures, note by (A.73) and (A.74) that b

(2π~)²

∞

X

j=0

Z

R

Z

R³

p²

ψ, P_u,p,j^~,b ψ

dudp= 1 2π~

Z

R

Z

R³

p²

g^~_u,p, g_u,p^~ dudp

= 1

~ Z

R

Z

R³

Z

R²

p² F₃

f^~(· −u)ψ

(y,~⁻¹p)

2dydudp

=~² Z

R

Z

R³

Z

R²

F₃

∂₃²(f^~(· −u)ψ) (y, p)

2dydudp

= Z

R³

f^~(· −u)ψ,−~²∂₃²(f^~(· −u)ψ) du.

Similarly, also using (A.73) and (A.74), and recalling (A.63), we get b

(2π~)²

∞

X

j=0

Z

R

Z

R³

~b(2j+ 1)

ψ, P_u,p,j^~,b ψ dudp

= ~ 2π

∞

X

j=0

Z

R

Z

R³

g^~_u,p, H_~⁻¹_bA^⊥Π⁽²⁾_j g_u,p^~ dudp

= ~ 2π

Z

R

Z

R³

g_u,p^~ , H_~⁻¹_bA^⊥g^~_u,p dudp

= Z

R³

f^~(· −u)ψ,~²(H_~⁻¹_bA^⊥⊗1L²(R))(f^~(· −u)ψ) du.

Since (−i~∇+bA)²=~²(H_~⁻¹_bA^⊥−∂₃²), combining these with (A.81) yields ψ,(−i~∇+bA)²ψ

=−~kψk²₂ Z

R³

(∇f(u))²du

+ b

(2π~)²

∞

X

j=0

Z

R

Z

R³

(p²+~b(2j+ 1))

ψ, P_u,p,j^~,b ψ dudp.

Now, since(σ·(−i~∇+bA))² = (−i~∇+bA)²1_C²+~bσ3, we get forΦ∈L²(R³;C²), Φ,(σ·(−i~∇+bA))²Φ

= b

(2π~)²

∞

X

j=0

Z

R

Z

R³

(p²+~b(2j+ 1))

Φ, P_u,p,j^~,b 1_C²Φ +~b

Φ, P_u,p,j^~,b σ3Φ dudp

−~kΦk²_L2(R³;C²)

Z

R³

(∇f(u))²du

= b

(2π~)² X

s=±1

∞

X

j=0

Z

R

Z

R³

(p²+~b(2j+ 1 +s))

Φ, P_u,p,j^~,b P_sΦ dudp

−~kΦk²_L2(R³;C²)

Z

R³

(∇f(u))²du.

Applying this to the first component of the wave functionΨwhile keeping all other variables fixed, we finally obtain

D Ψ,

N

X

j=1

(σ·(−i~∇_j+bA(xj)))²Ψ E

=N Z

(R³×{±1})^N−1

Ψ(·, z),(σ·(−i~∇+bA))²Ψ(·, z)

L²(R³;C²)dz

= b

(2π~)² X

s=±1

∞

X

j=0

Z

R

Z

R³

(p²+~b(2j+ 1 +s))N

Ψ, P_u,p,j^~,b P_sΨ dudp

−~N Z

R³

(∇f(u))²du, finishing the proof.

A.4.3 Limiting measures, strong magnetic fields

We now fix a real-valued, even and normalized function f ∈ L²(R³), along with a sequence (Ψ_N)_N≥1 of normalized functions with Ψ_N ∈ VN

L²(R³;C²) for each N. We investigate the measures m^(k)_f,Ψ

N in the limit as N tends to infinity, when β_N is a sequence with βN → β, 0 < β ≤ ∞. This corresponds to the regime where the distance between the Landau bands of the Pauli operator remains bounded from below.

Lemma A.34. For each k ≥ 1 there is a symmetric function m^(k)_f ∈ L¹(Ω^k)∩ L^∞(Ω^k) with 0 ≤m^(k)_f ≤1 such that, along a common (not displayed) subsequence in N,

Z

Ω^k

m^(k)_f,Ψ

N(ξ)ϕ(ξ) dξ −→

Z

Ω^k

m^(k)_f (ξ)ϕ(ξ) dξ (A.82) for all ϕ∈L¹(Ω^k) +L^∞_ε (Ω^k), as N tends to infinity.

The proof of this lemma is a standard exercise in functional analysis, using the boundedness of the sequence (m^(k)_f,Ψ

N)_N≥k both in L¹(Ω^k) and in L^∞(Ω^k), and we leave the details to the reader.

If the sequence of measures(m^(k)_f,Ψ

N)_N≥k istight, that is, if

R→∞lim lim sup

N→∞

Z

|ξ1|+···+|ξ_k|≥R

m^(k)_f,Ψ

N(ξ) dξ = 0.

then all the properties of the measures in Lemma A.31carry over to the limit, and the weak convergence in LemmaA.34is strengthened. We collect these observations in the lemma below, but the proof (which is elementary) will be omitted. The key ingredient for the proof is the fact that

Z

|ξ|≤R

m^(k)_f,Ψ

N(ξ) dξ −−−−→^R→∞

Z

Ω^k

m^(k)_f,Ψ

N(ξ) dξ= (2π~)^2k b^k

N! (N−k)!

uniformly inN asR tends to infinity, whenever(m^(k)_f,Ψ

N)N≥k is tight.

Lemma A.35. Suppose that (m⁽¹⁾_f,Ψ

N)N∈N is a tight sequence. Then we have (1) (m^(k)_f,Ψ

N)N≥k is also tight for each k≥1.

(2) The limit measures m^(k)_f are probability measures. More precisely, 1

(2π)^2k β^k (1 +β)^k

Z

Ω^k

m^(k)_f (ξ) dξ = 1. (A.83) (3) The compatibility relation (A.77) is preserved in the limit, that is, for k ≥ 2

and almost every ξ∈Ω^k−1, 1 (2π)²

β 1 +β

Z

Ω

m^(k)_f (ξ, ξk) dξk=m^(k−1)_f (ξ). (A.84) (4) The convergence in (A.82) holds on all of L¹(Ω^k) +L^∞(Ω^k).

We now formulate the de Finetti theorem which serves as the main abstract tool in our proof of the lower bound of the energy in Theorem A.5. The version of the theorem below is essentially [5, Theorem 2.6] For some additional details, see e.g.

[24].

Theorem A.36 (de Finetti). Let M ⊆Ωbe a locally compact subset, and m^(k)∈ L¹(M^k) a family of symmetric positive densities satisfying for some c > 0 and all k≥1 that 0≤m^(k)≤1, and

c Z

M

m^(k)(ξ₁, . . . , ξ_k) dξ_k=m^(k−1)(ξ₁, . . . , ξk−1)

with m⁽⁰⁾ = 1. Then there exists a unique Borel probability measure P on the set S =

µ∈L¹(M)

0≤µ≤1, c Z

M

µ(ξ) dξ= 1 such that for all k≥1, in the sense of measures,

m^(k)= Z

S

µ^⊗kdP(µ). (A.85)

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